---
title: "Temperature dependence of dipolar polarization"
subject: "Electrical Engineering Material"
chapter: 3
key: "semester_3/electrical_engineering_material:dipolar-polarization-temperature"
source: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/dipolar-polarization-temperature
site: BCT Engineering
---

# Temperature dependence of dipolar polarization

Electrical Engineering Material · Chapter 3: Dielectric Materials

- Asked in: 2076 Chaitra, 4 marks; 2073 Chaitra, 6 marks (asked 2 times)
- Video: [Frequency and temperature dependence of dielectric polarization | Dielectrics](https://www.youtube.com/watch?v=d7aEl0RjeF0) — Raheemsir jee Physics
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electrical_engineering_material/dipolar-polarization-temperature
- Whole chapter: https://www.bctengineer.com/md/semester_3/electrical_engineering_material/chapter-3

## Question

Show that dipolar polarization is a temperature dependent parameter.

## Short answer

For weak fields, average dipole moment ⟨p⟩ = p0²E/(3kT), giving dipolar polarizability α_d = p0²/(3kT). Thus α_d ∝ 1/T, decreasing with rising temperature because thermal agitation opposes alignment.

## Answer

**Dipolar (orientational) polarization** occurs in materials with permanent molecular dipoles $p_0$ (e.g. H₂O, HCl). Without a field the dipoles point randomly, so $P = 0$. An applied field tries to align them, while thermal agitation tries to keep them random. The result depends on temperature.

```text
 E = 0                E -->
  \  /  |  -           ->  /  ->  \
  -  \  /  |           ->  ->  /  ->
  random, P = 0        partly aligned, P > 0
```

#### Energy of a dipole in a field

A dipole at angle $\theta$ to the field $E$ has potential energy

$$
U = -p_0E\cos\theta
$$

and its component along the field is $p_0\cos\theta$.

#### Average dipole moment (Boltzmann statistics)

The probability of a dipole having orientation $\theta$ is proportional to $e^{-U/kT}$. The number of dipoles in the solid angle $d\Omega = 2\pi\sin\theta\,d\theta$ is proportional to $e^{p_0E\cos\theta/kT}\sin\theta\,d\theta$. So

$$
\langle p_0\cos\theta \rangle = p_0\frac{\int_0^{\pi}\cos\theta\, e^{a\cos\theta}\sin\theta\, d\theta}{\int_0^{\pi}e^{a\cos\theta}\sin\theta\, d\theta}, \qquad a = \frac{p_0E}{kT}
$$

Put $x = \cos\theta$, $dx = -\sin\theta\,d\theta$:

$$
\langle p_0\cos\theta \rangle = p_0\frac{\int_{-1}^{1}x e^{ax}dx}{\int_{-1}^{1}e^{ax}dx} = p_0\left[\coth a - \frac{1}{a}\right] = p_0 L(a)
$$

$L(a)$ is the **Langevin function**.

#### Usual condition (weak field, normal temperature)

Normally $p_0E \ll kT$ (e.g. $p_0 \sim 10^{-30}$ C m, $E = 10^{6}$ V/m gives $p_0E \approx 10^{-24}$ J, while $kT \approx 4 \times 10^{-21}$ J at 300 K), so $a \ll 1$ and $L(a) \approx a/3$. Then

$$
\langle p \rangle = \frac{p_0^2 E}{3kT}
$$

so the dipolar polarizability and polarization are

$$
\alpha_d = \frac{p_0^2}{3kT}, \qquad P_d = N\frac{p_0^2}{3kT}E
$$

#### Conclusion

$\alpha_d \propto 1/T$: as temperature rises, thermal agitation disorders the dipoles more and dipolar polarization falls. This is the **Debye relation**. Including electronic polarization, the total polarizability is

$$
\alpha = \alpha_e + \frac{p_0^2}{3kT}
$$

A plot of $\alpha$ (or $\varepsilon_r - 1$) against $1/T$ is a straight line: its slope gives $p_0$ and its intercept gives $\alpha_e$. For non-polar materials the slope is zero, which shows that only dipolar polarization depends on temperature.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The potential energy of a permanent dipole p0 at angle theta to field E is:
   A. U = p0 E sin(theta)
   B. U = -p0 E cos(theta)
   C. U = p0 E cos(theta)
   D. U = -p0 E tan(theta)
   Answer: B. Lowest energy occurs when the dipole is aligned with the field (theta = 0). (Key point: Dipole energy)

2. For p0 E much less than kT, the Langevin function L(a) is approximately:
   A. a
   B. 1
   C. a/3
   D. 3a
   Answer: C. For small a, coth a - 1/a is about a/3, giving average moment p0^2 E/(3kT). (Key point: Langevin approximation)

3. The dipolar polarizability is:
   A. alpha_d = 3kT/p0^2, increasing with T
   B. alpha_d = p0^2/(3kT), decreasing as 1/T
   C. alpha_d = p0^2/(3kT), independent of T
   D. alpha_d = p0 kT/3, increasing with T
   Answer: B. Thermal agitation randomizes the dipoles more at higher T, reducing alignment (Debye relation). (Key point: Temperature dependence)

4. In a plot of total alpha = alpha_e + p0^2/(3kT) versus 1/T, the intercept gives:
   A. p0
   B. alpha_e
   C. k
   D. N
   Answer: B. The slope gives p0^2/(3k) and the intercept is the temperature-independent alpha_e. (Key point: Plot of alpha versus 1/T)

## Related in Electrical Engineering Material

- [Temperature dependence of carrier concentration and mobility](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Atemperature-carrier-mobility) — Electrical Engineering Material, chapter 6, asked 2×
- [Intrinsic concentration and its temperature dependence](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Aintrinsic-concentration-derivation) — Electrical Engineering Material, chapter 6

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